Nie jem nabialu.(easy to read "nabialu" as it is) I do not
eat diary products (containing milk and egg products)

Nie jem produktow zwierzecych.(read "sweshenzikh" I do not
eat animal products (made of animals). Some may understand this
as if you do not eat milk,eggs products as well but not nrcesserly.They
will take you for an vegetarian.

PRONOUNCIATION
SZ like english SH as in "shovel"
J like eng. Y as in "yes"
C similar to eng. "S" (not quite but, it'll get you by)
W like eng. V
I like eng. "EE" as in "feed"

Polish word for vegan is wegan but it is still not recognized
by people. Being a vegetarian is still a surprise to many. They
may understand your request for not eating milk products because
some people don't for other health reasons. You must still explain
what you don't want on your plate.In this case the safest way is
to say specificly what you want.

note: Jestem jaroszem.Old Polish name for vegetarian.(read "yaroshem")That
means usualy that you do not eat any meat. Some may ask you if you
take the fish. Eggs and milk products are not taken to consideration
in this case.